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KS4 Mathematics. S6 Transformations. S6 Transformations. Contents. A. S6.2 Reflection. A. S6.3 Rotation. A. S6.1 Symmetry. S6.4 Translation. A. S6.5 Enlargement. A. S6.6 Combining transformations. A. Reflection symmetry.
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KS4 Mathematics S6 Transformations
S6 Transformations Contents • A S6.2 Reflection • A S6.3 Rotation • A S6.1 Symmetry S6.4 Translation • A S6.5 Enlargement • A S6.6 Combining transformations • A
Reflection symmetry If you can draw a line through a shape so that one half is the mirror image of the other then the shape has reflection or line symmetry. The mirror line is called a line of symmetry. one line of symmetry four lines of symmetry no lines of symmetry
Reflection symmetry How many lines of symmetry do the following designs have? one line of symmetry five lines of symmetry three lines of symmetry
Rotational symmetry An object has rotational symmetry if it fits exactly onto itself when it is turned about a point at its centre. The order of rotational symmetry is the number of times the object fits onto itself during a 360° turn. If the order of rotational symmetry is one, then the object has to be rotated through 360° before it fits onto itself again. Only objects that have rotational symmetry of two or more are said to have rotational symmetry. We can find the order of rotational symmetry using tracing paper.
Rotational symmetry What is the order of rotational symmetry for the following designs? Rotational symmetry order 4 Rotational symmetry order 3 Rotational symmetry order 5
Reflection symmetry in 3-D shapes Sometimes a 3-D shape can be divided into two symmetrical parts. What is the shaded area called? This shaded area is called a plane of symmetry.
Reflection symmetry in 3-D shapes How many planes of symmetry does a cube have? We can divide the cube into two symmetrical parts here.
Reflection symmetry in 3-D shapes We can draw the other eight planes of symmetry for a cube, as follows:
Reflection symmetry in 3-D shapes How many planes of symmetry does a cuboid have? A cuboid has three planes of symmetry.
Reflection symmetry in 3-D shapes A regular octahedron A regular pentagonal prism A sphere How many planes of symmetry do the following solids have? Explain why any right prism will always have at least one plane of symmetry.
Rotational symmetry in 3-D shapes Does a cube have rotational symmetry? We can draw a line through the centre of the cube, here. This line is called an axis of symmetry. What is the order of rotational symmetry about this axis? How many axes of symmetry does a cube have?
Rotational symmetry in 3-D shapes An octahedron A cuboid A tetrahedron How many axes of symmetry do each of the following shapes have? What is the order of rotational symmetry about each axis?
S6 Transformations Contents S6.1 Symmetry • A • A S6.3 Rotation • A S6.2 Reflection S6.4 Translation • A S6.5 Enlargement • A S6.6 Combining transformations • A
Reflection An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example, Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.
Reflecting shapes A B C D If we reflect the quadrilateral ABCD in a mirror line, we label the image quadrilateral A’B’C’D’. A’ B’ object image C’ D’ mirror line or axis of reflection The image is congruent to the original shape.
Reflecting shapes A A’ B B’ object image C C’ D D’ mirror line or axis of reflection If we draw a line from any point on the object to its image, the line forms a perpendicular bisector to the mirror line.
Finding a line of reflection Construct the line that reflects shape A onto its image A’. A’ This is the line of reflection. A • Draw lines from any two vertices to their images. • Mark on the mid-point of each line. • Draw a line through the mid points.
S6 Transformations Contents S6.1 Symmetry • A S6.2 Reflection • A • A S6.3 Rotation S6.4 Translation • A S6.5 Enlargement • A S6.6 Combining transformations • A
Describing a rotation A rotation occurs when an object is turned around a fixed point. To describe a rotation we need to know three things: • The angle of the rotation. For example, ½ turn = 180° ¼ turn = 90° ¾ turn = 270° • The direction of the rotation. For example, clockwise or anticlockwise. • The centre of rotation. This is the fixed point about which an object moves.
Rotating shapes If we rotate triangle ABC 90° clockwise about point O the following image is produced: B object 90° A A’ image B’ C C’ O A is mapped onto A’, B is mapped onto B’ and C is mapped onto C’. The image triangle A’B’C’ is congruent to triangle ABC.
Rotating shapes The centre of rotation can also be inside the shape. For example, 90° O Rotating this shape 90° anticlockwise about point O produces the following image.
Determining the direction of a rotation Sometimes the direction of the rotation is not given. If this is the case then we use the following rules: A positive rotation is an anticlockwise rotation. A negative rotation is an clockwise rotation. For example, A rotation of 60° = an anticlockwise rotation of 60° A rotation of –90° = an clockwise rotation of 90° Explain why a rotation of 120° is equivalent to a rotation of –240°.
Inverse rotations The inverse of a rotation maps the image that has been rotated back onto the original object. For example, the following shape is rotated 90° clockwise about point O. 90° O What is the inverse of this rotation? Either, a 90° rotation anticlockwise, or a 270° rotation clockwise.
Inverse rotations The inverse of any rotation is either • A rotation of the same size, about the same point, but in the opposite direction, or • A rotation in the same direction, about the same point, but such that the two rotations have a sum of 360°. What is the inverse of a –70° rotation? Either, a 70° rotation, or a –290° rotation.
Finding the centre of rotation Find the point about which A is rotated onto its image A’. A A A’ • Draw lines from any two vertices to their images. • Mark on the mid-point of each line. • Draw perpendicular lines from each of the mid-points. • The point where these lines meet is the centre of rotation.
Finding the angle of rotation Find the angle of rotation from A to its image A’. A A A’ 126° This is the angle of rotation • Join one vertex and its image to the centre of rotation. • Use a protractor to measure the angle of rotation.
S6 Transformations Contents S6.1 Symmetry • A S6.2 Reflection • A S6.3 Rotation • A S6.4 Translation • A S6.5 Enlargement • A S6.6 Combining transformations • A
Translation A’ image A A A A A A A A C’ B’ object object object object object object object object C C C C C C C C B B B B B B B B 5 We can describe this translation using the vector . 2 When an object is moved in a straight line in a given direction, we say that it has been translated. For example, we can translate triangle ABC 5 squares to the right and 2 squares up. Every point in the shape moves the same distance in the same direction.
Describing translations 3 For example, the vector describes a translation 3 right and 4 down. –4 When we describe a translation we always give the movement left or right first followed by the movement up or down. We can also describe translations using vectors. As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left. One more way of describing a translation is to give the direction as an angle and the distance as a length.
Describing translations When a shape is translated the image is congruent to the original. The orientations of the original shape and its image are the same. An inverse translation maps the image that has been translated back onto the original object. What is the inverse of a translation 7 units to the left and 3 units down? The inverse is an equal move in the opposite direction. That is, 7 units right and 3 units up.
Inverse translations What is the inverse of the translation ? 3 a –a The inverse of is . The inverse of the translation is –3 –3 –4 b –b 4 4 This vector translates the object 3 units to the left and 4 units up. The inverse of this translation is a movement 3 units to the right and 4 units down. In general,
S6 Transformations Contents S6.1 Symmetry • A S6.2 Reflection • A S6.3 Rotation • A S6.5 Enlargement S6.4 Translation • A • A S6.6 Combining transformations • A
Enlargement A’ A Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. We say that shape A has been enlarged by scale factor 2.
Enlargement A’C’ B’C’ A’B’ 6 12 9 AB BC AC 4 8 6 When a shape is enlarged, any length in the image divided by the corresponding length in the original shape (the object) is equal to the scale factor. A’ A 6 cm 4 cm 9 cm 6 cm B B’ 8 cm C 12 cm C’ = = = the scale factor = = = 1.5
Congruence and similarity Is an enlargement congruent to the original object? Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal. In an enlarged shape the corresponding angles are the same but the lengths are different. The object and its image are similar. Reflections, rotations and translations produce images that are congruent to the original shape. Enlargements produce images that are similar to the original shape.
Find the scale factor What is the scale factor for the following enlargements? A’ A Scale factor = 3
Find the scale factor What is the scale factor for the following enlargements? B’ B Scale factor = 2
Find the scale factor What is the scale factor for the following enlargements? C’ C Scale factor = 3.5
Find the scale factor What is the scale factor for the following enlargements? D D’ Scale factor = 0.5
